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Slice knots in \(S^ 3\). (English) Zbl 0542.57007

Let K be a knot in \(S^ 3\) and F a Seifert surface for K with Seifert pairing \(\theta\) : \(H_ 1(F)\times H_ 1(F)\to {\mathbb{Z}}\). Let \(\tau (K,\chi)\) be the Casson-Gordon invariant. The main theorem is: If K is a slice knot, then there is a direct summand H of \(H_ 1(F)\) such that \((1)\quad 2 rank H=rank H_ 1(F), (2)\quad \theta(H\times H)=0, (3)\quad \tau(K,\chi)=0\) for suitable \(\chi\). If K is a genus-one knot, some condition on the signature \(\sigma_{(s/m)}\), \(0<s<m\), is also obtained. Furthermore, for these knots, the author calculates \(\tau (K,\chi)\) explicitly.
Reviewer: K.Murasugi

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
11E16 General binary quadratic forms
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